Thomas Calculus 13th [Solutions]

i j k
Chapter 12 Practice Exercises 917
46. A vector in the direction of the planes normal is n u v 2 3 1 7i 3j 5k and P (1, 2, 3) on the
plane 7( x 1) 3( y 2) 5( z 3) 0 7x 3y 5z
14.
1 1 2
47. Yes; v n 2i 4j k 2i j 0k 2 2 4 1 1 0 0 the vector is orthogonal to the planes normal
v is parallel to the plane
48. n PP0 0 represents the half-space of points lying on one side of the plane in the direction which the normal
n points
i j k
49. A normal to the plane is n AB AC 2 0 1 i 2j 2k the distance is
2 1 0
4 2 2
d AP n i j i j k 1 8 0
n
1 4 4
3
3
50. P (0, 0, 0) lies on the plane 2x 3y 5z 0, and PS 2i 2j 3k with n 2i 3j 5k
d
n PS 4 6 15 25
| n| 4 9 25 38
i j k
51. n 2i j k is normal to the plane n v 2 1 1 0i 3j 3k 3j 3k is orthogonal to v and
1 1 1
parallel to the plane
52. The vector B C is normal to the plane of B and C A ( B C ) is orthogonal to A and parallel to the plane
of B and C:
i j k
i j k
B C 1 2 1 5i 3j k and A ( B C) 2 1 1 2i 3j k
1 1 2
5 3 1
A ( B C ) 4 9 1 14 and u 1
47
2i 3j k is the desired unit vector.
i j k
53. A vector parallel to the line of intersection is v n1 n2 1 2 1 5i j 3 k | v | 25 1 9 35
1 1 2
2 v 2
| v| 35
5 i j 3 k is the desired vector.
54. The line containing (0, 0, 0) normal to the plane is represented by x 2 t, y t , and z t . This line
intersects the plane 3x 5y 2z 6 when 3(2 t) 5( t) 2( t) 6 t 2 the point is 4 , 2 , 2 .
3
3 3 3
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